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# Writing Inequalities

Inequality is a statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions.

Symbols used are <, >, ≤, ≥

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One example of solving two-step inequalities is given below for understanding.

Writing inequalities is a fundamental skill in mathematics, often used to express relationships between quantities that are not necessarily equal. Here are some examples of how to write inequalities in various contexts:

Example 1: Simple Inequality

Let’s say you want to express the inequality “x is greater than 5”:

x > 5

This inequality states that the value of “x” is greater than 5.

To express “x increased by 3 is less than or equal to 10”:

x + 3 ≤ 10

This inequality indicates that when you add 3 to “x,” the result is less than or equal to 10.

Example 3: Inequality with Subtraction

To express “m decreased by 7 is greater than or equal to 15”:

m – 7 ≥ 15

This inequality indicates that when you subtract 7 from “m,” the result is greater than or equal to 15.

Example 4: Inequality with Multiplication

For the inequality “2 times y is greater than 16”:

2y > 16

Here, the inequality shows that the product of 2 and “y” is greater than 16.

Example 5: Inequality with Division

For the inequality  “9 divided by p is less than 27”:

9/p  < 27

Here, the inequality shows that the 9 and “p” quotients are less than 27.

Example 6: Compound Inequality

Suppose you want to write the compound inequality “x is greater than 3 and less than 8”:

3 < x < 8

This compound inequality indicates that “x” falls within the range between 3 and 8, excluding the endpoints.

Example 7: Inequality with Fractions

To express “a divided by 4 is greater than or equal to 14 divided by 8”:

a/4 ≥ 14/8

This inequality states that the result of dividing “a” by 4 is greater than or equal to 14 divided by 8.

Example 8: Inequality with Variables on Both Sides

If you want to express “2 times y minus 5 is less than 7 times y minus 10”:

2y + 5 < 7y – 10

Here, the inequality combines addition, multiplication, and subtraction involving a variable “y.”

Remember that inequalities can be combined with different mathematical operations and symbols to express relationships between values. Always consider the context of the problem and choose the appropriate symbols (<, >, ≤, ≥) and operations to accurately represent the given information.

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Example problems on inequalities

Example 1: ­3(6x + 2) ≥ 48

Solution:

­3(6x + 2) ≥ 48

-18x – 6 ≥ 48

=> -18x ≥ 54

=> x ≤ -3

Example 2: 2(x + 10) ≥ 55 – 3x

Solution:

2x + 20 ≥ 55 – 3x

=> 5x ≥ 35

=> x ≥ 7

Example 3: 4x2 + 12 ≥ 2x(2x + 2)

Solution:

4x2 + 12 ≥ 2x(2x + 2)

4x2 + 12 ≥ 4x2 + 4x

4x ≤ 12

=> x ≤ 3

Example 4: 15 < 4w – 5 < 19

Solution:

Adding 5 on all sides, we have

20 < 4w < 24

Dividing by 4 on all sides

5 < w < 6

Example 5:

Solution:

Multiplying each side by the LCM of the denominators of 4, 3 and 6 which is 12, we have

3(x + 2) – 4(2 – x) + 2(4x – 5) < 48

=> 3x + 6 – 8 + 4x + 8x – 10 < 48

=> 15x – 12 < 48

=>  15x < 60

=> x < 4

Example 6:

Write inequality for the following graphs

Solution:

(a) x < 1

(b) x  ≥ -1

(c) -2 ≤ x ≤ 2

## Order of Operations FAQS

#### What is an inequality?

An inequality is a mathematical statement that describes the relationship between two quantities that are not necessarily equal. It uses symbols such as “<” (less than), “>” (greater than), “≤” (less than or equal to), and “≥” (greater than or equal to) to represent the relationship.

#### How do I solve an inequality?

Solving an inequality involves determining the range of values that satisfy the inequality. You can use similar techniques as solving equations, but with some differences due to the inequality symbols. Common steps include isolating the variable and considering the direction of the inequality to determine if the boundary values are included or not.

#### Can I perform the same operations on inequalities as I do on equations?

Yes, you can perform many of the same operations on inequalities as you do on equations. However, when multiplying or dividing by a negative number, there would be a change in the direction of the inequality sign. Also, remember that multiplying or dividing by a variable expression requires considering its sign.

#### How do I graph inequalities?

Graphing inequalities involves representing the solution set on a number line or a coordinate plane. For linear inequalities (involving one variable), you can shade the region that satisfies the inequality. The solution region is the overlapping shaded area for systems of inequalities (multiple inequalities).

#### What is the difference between a solution and a solution set for an inequality?

A solution to an inequality is a specific value that makes the inequality true. The solution set is the collection of all values that satisfy the inequality. It may be a range of values rather than a single value.

#### How do I interpret compound inequalities?

Compound inequalities are formed by combining two separate inequalities using the words “and” or “or.” “And” corresponds to the intersection of solution sets, while “or” corresponds to their union. For example, “x > 3 and x < 8” represents values of “x” between 3 and 8.

#### Can I graph inequalities on a coordinate plane?

Yes, you can graph inequalities on a coordinate plane. For linear inequalities in two variables, such as “y > 2x + 3,” you can graph the corresponding boundary line (dashed or solid) and shade the region that satisfies the inequality. Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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